1
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Chen L, Wang Y, Liu J, Wang H. Coloured noise induces phenotypic diversity with energy dissipation. Biosystems 2022; 214:104648. [PMID: 35218875 DOI: 10.1016/j.biosystems.2022.104648] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2021] [Revised: 02/17/2022] [Accepted: 02/20/2022] [Indexed: 11/02/2022]
Abstract
Genes integrate many different sources of noise to adapt their survival strategy with energy costs, but how this noise impacts gene phenotype switching is not fully understood. Here, we refine a mechanistic model with multiplicative and additive coloured noise and analyse the influence of noise strength (NS) and autocorrelation time (AT) on gene phenotypic diversity. Different from white noise, we found that in the autocorrelation time-scale plane, increasing the multiplicative noise will broaden the bimodal region of the gene product, and additive noise will induce bimodal region drift from the lower level to the higher level, while the AT will promote this transition. Specifically, the effect of AT on gene expression is similar to a feedback loop; that is, the AT of multiplicative noise will elongate the mean first passage time (MFPT) from the low stable state to the high stable state, but it will reduce the MFPT from the high stable state to the low stable state, and the opposite is true for additive noise. Moreover, these transitions will violate the detailed equilibrium and then consume energy. By effective topology network reconstruction, we found that when the NS is small, the more obvious the bimodality is, the lower the energy dissipation; however, when the NS is large, it will consume more energy with a tendency for bimodality. The overall analysis implies that living organisms will utilize noise strength and its autocorrelation time for better survival in complex and fluctuating environments.
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Affiliation(s)
- Leiyan Chen
- School of Sciences, Hainan University, Haikou, 570228, Hainan, People's Republic of China
| | - Yan Wang
- Department of Neurology, The First Affiliated Hospital, University of South China, HengYang, 421001, Hunan, People's Republic of China
| | - Jinrong Liu
- School of Sciences, Hainan University, Haikou, 570228, Hainan, People's Republic of China
| | - Haohua Wang
- School of Sciences, Hainan University, Haikou, 570228, Hainan, People's Republic of China; Hainan University, Coll Forestry, Key Laboratory of Genetics & Germplasm Innovation Tropical Special Fo, Ministry of Education, Haikou, 570228, Hainan, People's Republic of China; Hainan University, Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province, Haikou, 570228, Hainan, People's Republic of China.
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2
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Huang GR, Saakian DB, Hu CK. Accurate analytic solution of chemical master equations for gene regulation networks in a single cell. Phys Rev E 2018; 97:012412. [PMID: 29448337 DOI: 10.1103/physreve.97.012412] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2017] [Indexed: 12/21/2022]
Abstract
Studying gene regulation networks in a single cell is an important, interesting, and hot research topic of molecular biology. Such process can be described by chemical master equations (CMEs). We propose a Hamilton-Jacobi equation method with finite-size corrections to solve such CMEs accurately at the intermediate region of switching, where switching rate is comparable to fast protein production rate. We applied this approach to a model of self-regulating proteins [H. Ge et al., Phys. Rev. Lett. 114, 078101 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.078101] and found that as a parameter related to inducer concentration increases the probability of protein production changes from unimodal to bimodal, then to unimodal, consistent with phenotype switching observed in a single cell.
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Affiliation(s)
- Guan-Rong Huang
- Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan
| | - David B Saakian
- Theoretical Physics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
| | - Chin-Kun Hu
- Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan.,Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.,Department of Systems Science, University of Shanghai for Science and Technology, Shanghai 200093, China.,Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan
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3
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Wang H, Liu P, Li Q, Zhou T. Entangled signal pathways can both control expression stability and induce stochastic focusing. FEBS Lett 2018; 592:1135-1149. [DOI: 10.1002/1873-3468.13012] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2017] [Revised: 01/28/2018] [Accepted: 02/08/2018] [Indexed: 11/08/2022]
Affiliation(s)
- Haohua Wang
- Department of Mathematics College of Information Science and Technology Hainan University Haikou China
| | - Peijiang Liu
- School of Statistics and Mathematics Guangdong University of Finance & Economics Guangzhou China
| | - Qingqing Li
- Guangdong Province Key Laboratory of Computational Science School of Mathematics Sun Yat‐Sen University Guangzhou China
| | - Tianshou Zhou
- Guangdong Province Key Laboratory of Computational Science School of Mathematics Sun Yat‐Sen University Guangzhou China
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4
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Kar P, Cherstvy AG, Metzler R. Acceleration of bursty multiprotein target search kinetics on DNA by colocalisation. Phys Chem Chem Phys 2018; 20:7931-7946. [DOI: 10.1039/c7cp06922g] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
Abstract
Proteins are capable of locating specific targets on DNA by employing a facilitated diffusion process with intermittent 1D and 3D search steps. We here uncover the implications of colocalisation of protein production and DNA binding sites via computer simulations.
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Affiliation(s)
- Prathitha Kar
- Dept of Inorganic and Physical Chemistry
- Indian Institute of Science
- Bengaluru
- India
- Institute for Physics & Astronomy
| | - Andrey G. Cherstvy
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
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5
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Feng S, Sáez M, Wiuf C, Feliu E, Soyer OS. Core signalling motif displaying multistability through multi-state enzymes. J R Soc Interface 2017; 13:rsif.2016.0524. [PMID: 27733693 PMCID: PMC5095215 DOI: 10.1098/rsif.2016.0524] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2016] [Accepted: 09/06/2016] [Indexed: 12/18/2022] Open
Abstract
Bistability, and more generally multistability, is a key system dynamics feature enabling decision-making and memory in cells. Deciphering the molecular determinants of multistability is thus crucial for a better understanding of cellular pathways and their (re)engineering in synthetic biology. Here, we show that a key motif found predominantly in eukaryotic signalling systems, namely a futile signalling cycle, can display bistability when featuring a two-state kinase. We provide necessary and sufficient mathematical conditions on the kinetic parameters of this motif that guarantee the existence of multiple steady states. These conditions foster the intuition that bistability arises as a consequence of competition between the two states of the kinase. Extending from this result, we find that increasing the number of kinase states linearly translates into an increase in the number of steady states in the system. These findings reveal, to our knowledge, a new mechanism for the generation of bistability and multistability in cellular signalling systems. Further the futile cycle featuring a two-state kinase is among the smallest bistable signalling motifs. We show that multi-state kinases and the described competition-based motif are part of several natural signalling systems and thereby could enable them to implement complex information processing through multistability. These results indicate that multi-state kinases in signalling systems are readily exploited by natural evolution and could equally be used by synthetic approaches for the generation of multistable information processing systems at the cellular level.
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Affiliation(s)
- Song Feng
- School of Life Sciences, University of Warwick, Coventry, UK
| | - Meritxell Sáez
- Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - Carsten Wiuf
- Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - Elisenda Feliu
- Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - Orkun S Soyer
- School of Life Sciences, University of Warwick, Coventry, UK
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6
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Determining the Limitations and Benefits of Noise in Gene Regulation and Signal Transduction through Single Cell, Microscopy-Based Analysis. J Mol Biol 2017; 429:1143-1154. [PMID: 28288800 DOI: 10.1016/j.jmb.2017.03.007] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2016] [Revised: 03/02/2017] [Accepted: 03/06/2017] [Indexed: 12/22/2022]
Abstract
Stochastic fluctuations, termed "noise," in the level of biological molecules can greatly impact cellular functions. While biological noise can sometimes be detrimental, recent studies have provided an increasing number of examples in which biological noise can be functionally beneficial. Rather than provide an exhaustive review of the growing literature in this field, in this review, we focus on single-cell studies based on quantitative microscopy that have generated a deeper understanding of the sources, characteristics, limitations, and benefits of biological noise. Specifically, we highlight studies showing how noise can help coordinate the expression of multiple downstream target genes, impact the channel capacity of signaling networks, and interact synergistically with oscillatory dynamics to enhance the sensitivity of signal processing. We conclude with a discussion of current challenges and future opportunities.
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7
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Liu P, Wang H, Huang L, Zhou T. The dynamic mechanism of noisy signal decoding in gene regulation. Sci Rep 2017; 7:42128. [PMID: 28176840 PMCID: PMC5296728 DOI: 10.1038/srep42128] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2016] [Accepted: 01/06/2017] [Indexed: 11/08/2022] Open
Abstract
Experimental evidence supports that signaling pathways can induce different dynamics of transcription factor (TF) activation, but how an input signal is encoded by such a dynamic, noisy TF and further decoded by downstream genes remains largely unclear. Here, using a system of stochastic transcription with signal regulation, we show that (1) keeping the intensity of the signal noise invariant but prolonging the signal duration can both enhance the mutual information (MI) and reduce the energetic cost (EC); (2) if the signal duration is fixed, the larger MI needs the larger EC, but if the signal period is fixed, there is an optimal time that the signal spends at one lower branch, such that MI reaches the maximum; (3) if both the period and the duration are simultaneously fixed, increasing the input noise can always enhance MI in the case of transcription regulation rather than in the case of degradation regulation. In addition, we find that the input noise can induce stochastic focusing in a regulation-dependent manner. These results reveal not only the dynamic mechanism of noisy signal decoding in gene regulation but also the essential role of external noise in controlling gene expression levels.
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Affiliation(s)
- Peijiang Liu
- School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
| | - Haohua Wang
- School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
- Department of Mathematics College of Information Science and Technology Hainan University, Haikou 570228, People’s Republic of China
| | - Lifang Huang
- School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, People’s Republic of China
| | - Tianshou Zhou
- School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
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8
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Firman T, Ghosh K. Competition enhances stochasticity in biochemical reactions. J Chem Phys 2014; 139:121915. [PMID: 24089727 DOI: 10.1063/1.4816527] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study stochastic dynamics of two competing complexation reactions (i) A + B↔AB and (ii) A + C↔AC. Such reactions are common in biology where different reactants compete for common resources--examples range from binding enzyme kinetics to gene expression. On the other hand, stochasticity is inherent in biological systems due to small copy numbers. We investigate the complex interplay between competition and stochasticity, using coupled complexation reactions as the model system. Within the master equation formalism, we compute the exact distribution of the number of complexes to analyze equilibrium fluctuations of several observables. Our study reveals that the presence of competition offered by one reaction (say A + C↔AC) can significantly enhance the fluctuation in the other (A + B↔AB). We provide detailed quantitative estimates of this enhanced fluctuation for different combinations of rate constants and numbers of reactant molecules that are typical in biology. We notice that fluctuations can be significant even when two of the reactant molecules (say B and C) are infinite in number, maintaining a fixed stoichiometry, while the other reactant (A) is finite. This is purely due to the coupling mediated via resource sharing and is in stark contrast to the single reaction scenario, where large numbers of one of the components ensure zero fluctuation. Our detailed analysis further highlights regions where numerical estimates of mass action solutions can differ from the actual averages. These observations indicate that averages can be a poor representation of the system, hence analysis that is purely based on averages such as mass action laws can be potentially misleading in such noisy biological systems. We believe that the exhaustive study presented here will provide qualitative and quantitative insights into the role of noise and its enhancement in the presence of competition that will be relevant in many biological settings.
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Affiliation(s)
- Taylor Firman
- Department of Physics and Astronomy, University of Denver, Denver, Colorado 80208, USA
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9
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Grima R, Schmidt DR, Newman TJ. Steady-state fluctuations of a genetic feedback loop: an exact solution. J Chem Phys 2012; 137:035104. [PMID: 22830733 DOI: 10.1063/1.4736721] [Citation(s) in RCA: 79] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution [J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Phys. Rev. E 72, 051907 (2005), and subsequent studies]. We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.
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Affiliation(s)
- R Grima
- SynthSys Edinburgh, School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JR, United Kingdom.
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10
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Qian H. Cooperativity in Cellular Biochemical Processes: Noise-Enhanced Sensitivity, Fluctuating Enzyme, Bistability with Nonlinear Feedback, and Other Mechanisms for Sigmoidal Responses. Annu Rev Biophys 2012; 41:179-204. [DOI: 10.1146/annurev-biophys-050511-102240] [Citation(s) in RCA: 71] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195;
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11
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Liu Y, Li N. Influences of a periodic signal on a noisy synthetic gene network. Sci China Chem 2011. [DOI: 10.1007/s11426-011-4285-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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12
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Daniel R, Almog R, Shacham-Diamand Y. Stochastic signaling in biochemical cascades and genetic systems in genetically engineered living cells. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041903. [PMID: 20481749 DOI: 10.1103/physreve.81.041903] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2009] [Revised: 10/19/2009] [Indexed: 05/29/2023]
Abstract
Living cells, either prokaryote or eukaryote, can be integrated within whole-cell biochips (WCBCs) for various applications. We investigate WCBCs where information is extracted from the cells via a cascade of biochemical reactions that involve gene expression. The overall biological signal is weak due to small sample volume, low intrinsic cell response, and extrinsic signal loss mechanisms. The low signal-to-noise ratio problem is aggravated during initial detection stages and limits the minimum detectable signal or, alternatively, the minimum detection time. Taking into account the stochastic nature of biochemical process, we find that the signal is accompanied by relatively large noise disturbances. In this work, we use genetically engineered microbe sensors as a model to study the biochips output signal stochastic behavior. In our model, the microbes are designed to express detectable reporter proteins under external induction. We present analytical approximated expressions and numerical simulations evaluating the fluctuations of the synthesized reporter proteins population based on a set of equations modeling a cascade of biochemical and genetic reactions. We assume that the reporter proteins decay more slowly than messenger RNA molecules. We calculate the relation between the noise of the input signal (extrinsic noise) and biochemical reaction statistics (intrinsic noise). We discuss in further details two cases: (1) a cascade with large decay rates of all biochemical reactions compared to the protein decay rate. We show that in this case, the noise amplitude has a positive linear correlation with the number of stages in the cascade. (2) A cascade which includes a stable enzymatic-binding reaction with slow decay rate. We show that in this case, the noise strongly depends on the protein decay rate. Finally, a general observation is presented stating that the noise in whole-cell biochip sensors is determined mainly by the first reactions in the genetic system with weak dependence on the number of stages in the cascade.
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Affiliation(s)
- Ramiz Daniel
- Department of Physical Electronic, Electrical Engineering Faculty, Tel Aviv University, Ramat Aviv 69978, Israel.
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13
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Gerstung M, Timmer J, Fleck C. Noisy signaling through promoter logic gates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:011923. [PMID: 19257085 DOI: 10.1103/physreve.79.011923] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2007] [Revised: 10/24/2008] [Indexed: 05/27/2023]
Abstract
We study the influence of noisy transcription factor signals on cis-regulatory promoter elements. These elements process the probability of binary binding events analogous to computer logic gates. At equilibrium, this probability is given by the so-called input function. We show that transcription factor noise causes deviations from the equilibrium value due to the nonlinearity of the input function. For a single binding site, the correction is always negative resulting in an occupancy below the mean-field level. Yet for more complex promoters it depends on the correlation of the transcription factor signals and the geometry of the input function. We present explicit solutions for the basic types of AND and OR gates. The correction size varies among these different types of gates and signal types, mainly being larger in AND gates and for correlated fluctuations. In all cases we find excellent agreement between the analytical results and numerical simulations. We also study the E. coli Lac operon as an example of an AND NOR gate. We present a consistent mathematical method that allows one to separate different sources of noise and quantifies their effect on promoter occupation. A surprising result of our analysis is that Poissonian molecular fluctuations, in contrast to external fluctuations, do no contribute to the correction.
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Affiliation(s)
- Moritz Gerstung
- Department of Biosystems Science and Engineering, ETH Zurich, Mattenstrasse 26, 4058 Basel, Switzerland.
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14
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Determining biological noise via single cell analysis. Anal Bioanal Chem 2008; 393:73-80. [PMID: 18958456 DOI: 10.1007/s00216-008-2431-z] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2008] [Revised: 09/16/2008] [Accepted: 09/23/2008] [Indexed: 10/21/2022]
Abstract
Single cell analysis techniques describe the cellular heterogeneity that originates from fundamental stochastic variations in each of the molecular processes underlying cell function. The quantitative description of this set of variations is called biological noise and includes intrinsic and extrinsic noise. The former refers to stochastic variations directly involved with a given process, while the latter is due to environmental factors associated with other processes. Mathematical models are successful in predicting noise trends in simple biological systems, but it takes single cell techniques such as flow cytometry and time lapse microscopy to determine and dissect biological noise. This review describes several approaches that have been successfully used to describe biological noise.
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15
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Sánchez Á, Kondev J. Transcriptional control of noise in gene expression. Proc Natl Acad Sci U S A 2008; 105:5081-6. [PMID: 18353986 PMCID: PMC2278180 DOI: 10.1073/pnas.0707904105] [Citation(s) in RCA: 126] [Impact Index Per Article: 7.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2007] [Indexed: 11/18/2022] Open
Abstract
Cis-regulatory control of transcription is the dominant form of regulation of gene expression. Recent experimental results suggest that, in addition to the mean expression level, cell-to-cell variability might also be transcriptionally regulated. Here, we develop a stochastic model of transcriptional regulation that allows us to calculate closed-form analytical expressions for the mean and variance of the protein and mRNA distributions for an arbitrarily complex cis-regulatory motif. Our model allows us to investigate how noise may be transcriptionally regulated independently from the mean expression. We show that our approach is in excellent agreement with stochastic simulations and experiment, and leads to an experimentally testable formula for the noise in gene expression as a function of inducer-molecule concentrations.
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Affiliation(s)
- Álvaro Sánchez
- Graduate Program in Biophysics and Structural Biology and
| | - Jané Kondev
- Department of Physics, Brandeis University, Waltham, MA 02454
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16
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Lan Y, Wolynes PG, Papoian GA. A variational approach to the stochastic aspects of cellular signal transduction. J Chem Phys 2007; 125:124106. [PMID: 17014165 DOI: 10.1063/1.2353835] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Cellular signaling networks have evolved to cope with intrinsic fluctuations, coming from the small numbers of constituents, and the environmental noise. Stochastic chemical kinetics equations govern the way biochemical networks process noisy signals. The essential difficulty associated with the master equation approach to solving the stochastic chemical kinetics problem is the enormous number of ordinary differential equations involved. In this work, we show how to achieve tremendous reduction in the dimensionality of specific reaction cascade dynamics by solving variationally an equivalent quantum field theoretic formulation of stochastic chemical kinetics. The present formulation avoids cumbersome commutator computations in the derivation of evolution equations, making the physical significance of the variational method more transparent. We propose novel time-dependent basis functions which work well over a wide range of rate parameters. We apply the new basis functions to describe stochastic signaling in several enzymatic cascades and compare the results so obtained with those from alternative solution techniques. The variational Ansatz gives probability distributions that agree well with the exact ones, even when fluctuations are large and discreteness and nonlinearity are important. A numerical implementation of our technique is many orders of magnitude more efficient computationally compared with the traditional Monte Carlo simulation algorithms or the Langevin simulations.
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Affiliation(s)
- Yueheng Lan
- Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3290, USA
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17
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Lan Y, Papoian GA. Evolution of complex probability distributions in enzyme cascades. J Theor Biol 2007; 248:537-45. [PMID: 17631318 DOI: 10.1016/j.jtbi.2007.06.008] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2007] [Revised: 05/29/2007] [Accepted: 06/06/2007] [Indexed: 10/23/2022]
Abstract
Unusual probability distribution profiles, including transient multi-peak distributions, have been observed in computer simulations of cell signaling dynamics. The emergence of these complex distributions cannot be explained using either deterministic chemical kinetics or simple Gaussian noise approximation. To develop physical insights into the origin of complex distributions in stochastic cell signaling, we compared our approximate analytical solutions of signaling dynamics with the exact numerical simulations. Our results are based on studying signaling in 2-step and 3-step enzyme amplification cascades that are among the most common building blocks of cellular protein signaling networks. We have found that while the multi-peak distributions are typically transient, and eventually evolve into single peak distributions, in certain cases these distributions may be stable in the limit of long times. We also have shown that introducing positive feedback loops results in diminution of the probability distribution complexity.
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Affiliation(s)
- Yueheng Lan
- Department of Chemistry, University of North Carolina, Chapel Hill, USA
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18
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19
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Levine J, Kueh HY, Mirny L. Intrinsic fluctuations, robustness, and tunability in signaling cycles. Biophys J 2007; 92:4473-81. [PMID: 17400695 PMCID: PMC1877790 DOI: 10.1529/biophysj.106.088856] [Citation(s) in RCA: 43] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Covalent modification cycles (e.g., phosphorylation-dephosphorylation) underlie most cellular signaling and control processes. Low molecular copy number, arising from compartmental segregation and slow diffusion between compartments, potentially renders these cycles vulnerable to intrinsic chemical fluctuations. How can a cell operate reliably in the presence of this inherent stochasticity? How do changes in extrinsic parameters lead to variability of response? Can cells exploit these parameters to tune cycles to different ranges of stimuli? We study the dynamics of an isolated phosphorylation cycle. Our model shows that the cycle transmits information reliably if it is tuned to an optimal parameter range, despite intrinsic fluctuations and even for small input signal amplitudes. At the same time, the cycle is sensitive to changes in the concentration and activity of kinases and phosphatases. This sensitivity can lead to significant cell-to-cell response variability. It also provides a mechanism to tune the cycle to transmit signals in various amplitude ranges. Our results show that signaling cycles possess a surprising combination of robustness and tunability. This combination makes them ubiquitous in eukaryotic signaling, optimizing signaling in the presence of fluctuations using their inherent flexibility. On the other hand, cycles tuned to suppress intrinsic fluctuations can be vulnerable to changes in the number and activity of kinases and phosphatases. Such trade-offs in robustness to intrinsic and extrinsic fluctuations can influence the evolution of signaling cascades, making them the weakest links in cellular circuits.
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Affiliation(s)
- Joseph Levine
- Computation and Neural Systems Option, California Institute of Technology, Pasadena, California, USA
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20
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Tao Y, Zheng X, Sun Y. Effect of feedback regulation on stochastic gene expression. J Theor Biol 2007; 247:827-36. [PMID: 17507034 DOI: 10.1016/j.jtbi.2007.03.024] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2007] [Revised: 03/10/2007] [Accepted: 03/18/2007] [Indexed: 11/19/2022]
Abstract
Stochastic noise in gene expression arises as a result of species in small copy number undergoing transitions between discrete chemical states. Here the noise in a single gene network is investigated using the Omega-expansion techniques. We show that the linear noise approximation implies an invariant relationship between the normalized variances and normalized covariance in steady-state statistics. This invariant relationship provides an exactly statistical interpretation for why the stochastic noise in gene expression should be measured by the normalized variance. The nature of the normalized variance reveals the basic relationship between the stochasticity and system size in gene expression. The linear noise approximation implies also that for both mRNA and protein, the total noise can be decomposed into two basic components, one concerns the contribution of average number of molecules, and other the contribution of interactions between mRNA and protein. For the situation with linear feedback, our results clearly show that for two genes with the same average number of protein molecules, the gene with negative feedback will have a small protein noise, i.e., the negative feedback will reduce the protein noise. For the effect of the burst size on the protein noise, we show also that the protein intrinsic noise will decrease with the increase of the burst size, but the protein extrinsic noise is independent of the burst size.
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Affiliation(s)
- Yi Tao
- Key Laboratory of Animal Ecology and Conservation Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing 100080, PR China.
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Xu BL, Tao Y. External noise and feedback regulation: steady-state statistics of auto-regulatory genetic network. J Theor Biol 2006; 243:214-21. [PMID: 16890960 DOI: 10.1016/j.jtbi.2006.06.003] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/26/2005] [Revised: 06/01/2006] [Accepted: 06/02/2006] [Indexed: 10/24/2022]
Abstract
The steady-state statistics of a single gene auto-regulatory genetic network with the additive external Gaussian white noises is investigated. The main result shows that the negative feedback will result in that the mRNA noise has a positive contribution to the protein noise, but the positive feedback will result in that the mRNA noise has a negative contribution to the protein noise. If there is no feed back, then the contribution of mRNA noise to protein noise is always positive. On the other hand, the analysis and numerical simulations of linear and nonlinear feedback show that it is possible that the negative feedback increases, but the positive feedback decreases, the protein noise.
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Affiliation(s)
- Bing-Liang Xu
- College of Grassland Sciences, Gansu Agriculture University, Lanzhou, PR China
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22
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Murugan R. Stochastic transcription initiation: Time dependent transcription rates. Biophys Chem 2006; 121:51-6. [PMID: 16442697 DOI: 10.1016/j.bpc.2005.12.010] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2005] [Revised: 12/21/2005] [Accepted: 12/21/2005] [Indexed: 10/25/2022]
Abstract
The noise in the central process such as transcription, replication and translation of the genomic DNA is very important since it can directly affect the phenotypic and behavioral aspects of an organism as well as the entire cellular function. Here we develop a model on the transcription process based on the assumption that the initiation of the transcription is a stochastic event and the transcription rates may be time dependent random quantities. We derive the central measure properties i.e. mean and the variance, of the distribution of the transcription rates. Our results show that the Fano factor which is a measure of deviation from the Poisson distribution associated with the fluctuations in the number of mRNA molecules deviates from unity due to the randomness in the transcription rates. However when the RNA polymerase molecule searches for the promoter sequences on the DNA lattice by random jumps, the Fano factor approaches the Poisson limit as the jump size associated with the RNA polymerase increases. Since the jump size associated with dynamics of RNAP molecule is positively correlated with the degree of super coiling of DNA, we argue that the super coiled or close-packed structure of DNA might have evolved to keep the noises at the transcriptional level in a minimum.
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Affiliation(s)
- R Murugan
- Department of Chemical Sciences, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India.
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23
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Abstract
Recently, several theoretical and experimental studies have been undertaken to probe the effect of stochasticity on gene expression (GE). In experiments, the GE response to an inducing signal in a cell, measured by the amount of mRNAs/proteins synthesized, is found to be either graded or binary. The latter type of response gives rise to a bimodal distribution in protein levels in an ensemble of cells. One possible origin of binary response is cellular bistability achieved through positive feedback or autoregulation. In this paper, we study a simple, stochastic model of GE and show that the origin of binary response lies exclusively in stochasticity. The transitions between the active and inactive states of the gene are random in nature. Graded and binary responses occur in the model depending on the relative stability of the activated and deactivated gene states with respect to that of mRNAs/proteins. The theoretical results on binary response provide a good description of the 'all-or-none' phenomenon observed in an eukaryotic system.
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Affiliation(s)
- Rajesh Karmakar
- Department of Physics, Bose Institute, 93/1, Acharya Prafulla Chandra Road, Kolkata-700 009, India
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24
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Tao Y, Jia Y, Dewey TG. Stochastic fluctuations in gene expression far from equilibrium: Ω expansion and linear noise approximation. J Chem Phys 2005; 122:124108. [PMID: 15836370 DOI: 10.1063/1.1870874] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The Omega expansion of the master equation is used to investigate the intrinsic noise in an autoregulatory gene expression system. This Omega expansion provides a mesoscale description of the system and is used to analyze the effect of feedback regulation on intrinsic noise when the system state is far from equilibrium. Using the linear noise approximation, analytic results are obtained for a single gene system with linear feedback that is far from equilibrium. Additionally, analytic expressions are obtained for nonlinear systems near equilibrium. Simulations of such autoregulatory reaction schemes with nonlinear feedback show that during the approach to equilibrium the noise is not always reduced by the strength of the feedback. This is contrary to results seen in the equilibrium limit which show decreased noise with feedback strength. These results demonstrate that the behavior of linearized systems near equilibrium cannot be readily applied to systems far from equilibrium and highlight the need to explore nonequilibrium domains in mesoscopic systems.
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Affiliation(s)
- Yi Tao
- Keck Graduate Institute of Applied Life Sciences, Claremont, California 91711, USA
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25
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26
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Tao Y. Intrinsic noise, gene regulation and steady-state statistics in a two-gene network. J Theor Biol 2004; 231:563-8. [PMID: 15488533 DOI: 10.1016/j.jtbi.2004.07.012] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2004] [Revised: 07/11/2004] [Accepted: 07/12/2004] [Indexed: 10/26/2022]
Abstract
The intrinsic noise in a two-gene network model is analysed. The technique of the Fokker-Planck approximation is used to investigate the statistics of noise when the system state is near a stable equilibrium. This is called also the steady-state statistics. The relative size of noise is measured by the Fano factor that is defined as the ratio of the variance to the mean. Our main result shows that in general, the noise control in a two-gene network might be a very complicated process, but for the repressor-repressor system that is a very important case in investigating the genetic switch, the relative size of noise, i.e. the Fano factor, must be bigger than one for both the repressor proteins.
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Affiliation(s)
- Yi Tao
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ont., Canada N2L 3C5.
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27
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Tao Y. Intrinsic and external noise in an auto-regulatory genetic network. J Theor Biol 2004; 229:147-56. [PMID: 15207470 DOI: 10.1016/j.jtbi.2004.03.011] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2003] [Revised: 02/24/2004] [Accepted: 03/12/2004] [Indexed: 10/26/2022]
Abstract
A single gene auto-regulatory network is analysed. The main goal is to investigate the effects of the negative and positive feedbacks on the intrinsic and external noises. The central finding of this paper is that: for the intrinsic noise, both the negative and positive feedback regulations increase the fluctuation strength of mRNA levels (where the fluctuation strength is measured by the Fano factor for both the fluctuations of mRNAs and proteins), and the negative feedback decreases, but the positive feedback increases, the fluctuation strength of proteins; for the external noise, the negative feedback not only increase the fluctuation strength of mRNA levels but also the fluctuation strength of proteins, and though the effect of the positive feedback on the fluctuation strength of mRNA levels depends on the size of positive feedback parameter k, the positive feedback must decrease the fluctuation strength of proteins.
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Affiliation(s)
- Yi Tao
- Centre for Structural and Functional Genomics, Concordia University, Montreal, Quebec, Canada H3G 3J7.
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28
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Abstract
Cells are intrinsically noisy biochemical reactors: low reactant numbers can lead to significant statistical fluctuations in molecule numbers and reaction rates. Here we use an analytic model to investigate the emergent noise properties of genetic systems. We find for a single gene that noise is essentially determined at the translational level, and that the mean and variance of protein concentration can be independently controlled. The noise strength immediately following single gene induction is almost twice the final steady-state value. We find that fluctuations in the concentrations of a regulatory protein can propagate through a genetic cascade; translational noise control could explain the inefficient translation rates observed for genes encoding such regulatory proteins. For an autoregulatory protein, we demonstrate that negative feedback efficiently decreases system noise. The model can be used to predict the noise characteristics of networks of arbitrary connectivity. The general procedure is further illustrated for an autocatalytic protein and a bistable genetic switch. The analysis of intrinsic noise reveals biological roles of gene network structures and can lead to a deeper understanding of their evolutionary origin.
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Affiliation(s)
- M Thattai
- Department of Physics, Room 13-2010, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
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